In the infinite-dimensional setting synchronous strategies give rise to commuting operator strategies: as shown in [1]}, for every synchronous strategy \({S}= (\tau ,\lbrace M^x\rbrace )\)  with Hilbert space \(\mathcal {H}\)  there exist another Hilbert space \(\mathcal {H}^{\prime }\) , a state \({|\psi \rangle } \in \mathcal {H}^{\prime }\) , and measurements \(\lbrace A^x\rbrace , \lbrace B^x\rbrace \)  on \(\mathcal {H}^{\prime }\)  for the provers respectively such that for all \(x,y \in \mathcal {X}\)  and \(a,b\in \mathcal {A}\) , the operators \(A^x_a\)  and \(B^y_b\)  commute and we have
\(\tau (M^x_a \, M^y_b) = {\langle \psi |} A^x_a \, B^y_b {|\psi \rangle }~.\) 
